Real estate math demystified

CHAPTER 1 Review of Math Skills 7 CHAPTER 2 Fractions, Decimals, and Percentages 20 CHAPTER 3 Commissions, Growth Rates, and CHAPTER 4 Legal Descriptions and Lot Size 45 CHAPTER 6 Time V

REAL ESTATE MATH

DEMYSTIFIED

Steven P Mooney

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CHAPTER 1 Review of Math Skills 7

CHAPTER 2 Fractions, Decimals, and Percentages 20

CHAPTER 3 Commissions, Growth Rates, and

CHAPTER 4 Legal Descriptions and Lot Size 45

CHAPTER 6 Time Value of Money (TVM) 64

iii

iv

CHAPTER 7 Mortgage Calculations Using TVM 87

CHAPTER 8 Appreciation and Depreciation 114

CHAPTER 9 The Closing and Closing Statements 129

CHAPTER 10 Real Estate Appraisal 145

CHAPTER 11 Real Estate Investment Analysis 174

CHAPTER 12 Risk in Real Estate 202

I would like to thank my wife Kate, for encouraging me to pursue an academiccareer in the late nineteen-seventies I would also like to thank Jack Friedmanfor providing me a great role model during my doctoral program at Texas A&MUniversity Jack also recommended Kate as a potential author to Grace Freedson,who now serves as our agent, whom I would also like to thank I would like tothank Julie Clasen for her hard work in creating many of the diagrams and listsused in the book.

Finally, I would like to thank every student I have ever taught at St Cloud StateUniversity since it was our interaction that provided me with the material for thisbook

v

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Other Titles in the Demystified Series

Advanced Calculus Demystified Home Networking Demystified Advanced Physics Demystfied Investing Demystified

Advanced Statistics Demystified Italian Demystified

Algebra Demystified Java Demystified

Anatomy Demystified JavaScript Demystified

Astronomy Demystified Macroeconomics Demystified Biology Demystified Math Proofs Demystified

Business Statistics Demystified Math Word Problems Demystified

Calculus Demystified OOP Demystified

Chemistry Demystified Options Demystified

Circuit Analysis Demystified Personal Computing Demystified College Algebra Demystified Physics Demystified

Databases Demystified Physiology Demystified

Data Structures Demystified Pre-Algebra Demystified

Diabetes Demystified Precalculus Demystified

Differential Equations Demystified Probability Demystified

Digital Electronics Demystified Project Management Demystified Earth Science Demystified Quantum Mechanics Demystified Electricity Demystified Relativity Demystified

Electronics Demystified Robotics Demystified

Environmental Science Demystified Six Sigma Demystified

Everyday Math Demystified Six Sigma Lite Demystified

Fertility Demystified Spanish Demystified

Financial Planning Demystified Statistics Demystified

French Demystified Thermodynamics Demystified Geometry Demystified Trigonometry Demystified

German Demystified Vitamins and Minerals Demystified

Real estate math is scary for some people This does not have to be the case In this

book we revert back to some math principles we mastered in elementary school

In addition we brush up on some calculations we performed in junior high school

and high school Finally we address some topics that are usually discovered in

college-level real estate courses This book is not designed for people who have

already taken those college-level courses; it is designed for the students who have

forgotten some (or most) of the math they took as a student in the past, but

are sincere in their desire to be able to analyze real estate from all the different

viewpoints This includes the viewpoint of the buyer, the seller, the lender, the

appraiser, and the investor I would certainly not discourage those who have taken

college-level courses from reading the book, however It will be an excellent

review of some topics and may well introduce a couple of new ones to you

This book is targeted at a broad group of readers It will certainly be beneficial

for those who are interested in obtaining their real estate broker’s or salesperson’s

license The individual investor would also be well advised to read this book

1

Copyright © 2007 by McGraw-Hill, Inc Click here for terms of use

2

Real estate students at the community college and university level would alsoprofit from reading this book Since I have been all of these people myself—areal estate salesperson and broker, a real estate investor, and a real estate student—

I understand the needs of each of these groups of people Now, as a real estateprofessor, I see the need for a book such as this to benefit the groups of peoplethat I have belonged to

There is much that is exciting about real estate analysis, and much of it isdependent on a firm math background If you don’t have that background rightnow, don’t worry You will have it by the time you finish this book I haveattacked this book not from the standpoint of a math expert I am very much

a practitioner when it comes to math, not a theory guy In order for math

to make sense to me, I have to be able to use it for something If you areinterested in math theory, you’d better find another book If you are interested

in how math can help you analyze real estate investments, find a quiet place,some paper, a pencil, and a financial calculator Then hang on; it’s going to bewild ride

Review

You will find that this book is divided into 13 chapters that range from simplereview to challenging new material The first chapter is a review of math skillsand addresses such concepts as equations and units of measure We discuss whatmakes an equation an equation and also some relevant units of measure that areused in real estate We close the chapter with a brief discussion of the financialcalculator and how that will be used in future chapters

Parts of a Whole

The second chapter addresses the concepts of fractions, decimals, and percentages.The manipulation of fractions, including the addition, subtraction, multiplicationand division of those fractions is discussed in detail This includes the manipu-lation of proper fractions, improper fractions, and mixed numbers Later in thechapter we look at converting those fractions to decimals and percentages Thesepercentages are important since much of real estate math is dealing with com-mission rates, rates of return, interest rates, and discount rates, all of which arepercentages

In Chapter 3 the topics of commission and growth rates are studied This is an

extension of the percentage discussion that begins at the end of Chapter 2 In

addi-tion to commission rates, which are of importance to buyers, sellers, brokers, and

salespeople, growth rates are discussed Both average growth rates and compound

growth rates are analyzed If you don’t know the difference between average and

compound growth rates, you need me; buy the book If you do know the

differ-ence between the two but you don’t know how to calculate them, you need me;

buy the book

Big Backyard?

The fourth chapter is called “Legal Descriptions and Lot Size: Does This Drawing

Make My Backyard Look Big?” There are three types of legal descriptions that

are the most commonly used in the United States They are the metes and bounds

system, the government survey system, and the subdivision plat system The

first one dates back to medieval England; the second, back to the days of the

Louisiana Purchase; and the third, to relatively more recent times In addition to

these systems of describing land, we use some of the tools from the first chapter

to calculate the area of a lot and the volume of a structure

Real Estate Taxes

Chapter 5 is all about real estate taxes and covers such topics as ad valorem

taxes, assessed value, estimated market values, mill rates, and state deed taxes

We address the concept of how value is determined for tax purposes We look at

the three approaches for appraising property, although in a much more cursory

way than in Chapter 10 on real estate appraisal Three different types of tax rates

are analyzed—the nominal tax rate, the average tax rate, and the effective tax rate

Time Value of Money

In Chapter 6, “Time Value of Money,” we start to hit the mother lode of the book

As I tell my students, this stuff isn’t important It’s critical! Once you understand

4

time value of money, you understand finance, you understand investments, youunderstand appraisal, and you understand life We will learn, mark, and inwardlydigest the concepts of present value, future value, ordinary annuity, annuity due,net present value, internal rate of return, sinking fund payment, and payment toamortize Your financial calculator will get its initial workout in this chapter Don’tworry if you don’t understand your financial calculator yet By the time you aredone with this book, you will be playing that instrument like Johann SebastianBach played the organ

Mortgages

We continue using the time value of money techniques when we move intothe mortgage calculations in Chapter 7 Mortgage payment calculations andprincipal balance calculations will become second nature after you finish read-ing this chapter and working the associated problems The impact of discountpoints and other prepaid items on loan yields are focused on Those yields areimportant both from the lender’s point of view and from the borrower’s point

of view Loan-to-value ratios are discussed, and homeowner’s equity is lated Finally, borrower qualifications and mortgage underwriting problems aretackled

calcu-Appreciation and Depreciation

Chapter 8 is “Appreciation and Depreciation: You Win Some, and You LoseSome.” The increase in property value over time is what we refer to as apprecia-tion The impact of appreciation on the cost of owning is one of the major factorsleading over 67 percent of the population in the United States to own rather thanrent their personal residence We will look at estimating the value of a residencegiven a certain rate of appreciation On the other end of changes in value is theconcept of depreciation Depreciation in two forms are addressed in the chapter.First of all the depreciation as it relates to real estate appraisal is discussed Thisform of depreciation is a loss in value due to any cause, and it comes into play inthe cost approach of the appraisal process The second form of depreciation that

we look at is the depreciation for tax purposes that is an important element in realestate investment analysis

Closing Statements

The ninth chapter is about closings and closing statements The closing statement

is the actual accounting of the purchase/sales transaction It tracks the expenses

and the prepayments associated with the transaction As much as it gives me

the willies to talk about accounting (my wife is an accounting professor), in the

closing statements we track both the buyer’s and the seller’s debits and credits

The bottom line in the buyer’s closing statement is how much additional cash they

have to bring to the closing The corresponding line on the seller’s statement will

tell the seller how much money he or she will be taking home from the closing

Topics de jour include expenses, prepaid items, accrued items, and prorated items

Appraisal

Real estate appraisal is the next topic in the book, and it appears in Chapter 10

The math portion of the sales comparison approach and the cost approach are

pretty much a function of adding and subtracting I feel confident you didn’t

buy this book in order to review adding and subtracting whole numbers and

dollars and cents That being the case, I focus most of my time and energies

in this chapter on the different forms of the income approach These include

the gross rent multiplier, the income capitalization, and the discounted cash flow

methods Within the discussion of the income capitalization method are methods

of calculating the capitalization rate in two different ways One is through market

extraction, and the other is called the band of investment method Hopefully, when

we finish up on the band of investment method, you will respond the same way

I did when I first saw it Wow! That is what I said, and that is what my students

say, well the bright ones, anyway

Real Estate Investment

An investor will only pay the present value of all cash flows to be received,

discounted at the required rate of return This is pretty much the heart of

Chapter 11 It draws extensively upon Chapter 6, “Time Value of Money,”

and Chapter 10, “Real Estate Appraisal.” The topics focused on are cash flow

estimation, net present value, and internal rate of return as they apply to

6

real estate investment Once you have completed this chapter, you will be able

to use these tools to analyze a student rental investment, a single-family homerental, a 4-unit building, a 5,000-unit apartment complex, or the Mall of America.The same concepts and analyses apply to all types of property

Risk

Risk is, in effect, the probability of achieving an undesired outcome From aninvestor’s viewpoint that would be achieving a rate of return that is lower thanthe expected rate of return For some reason investors don’t mind achieving arate of return higher than was expected Rather, it is the downside risk they areconcerned with In Chapter 12 we take a close look at the impact that risk has

on the investor, the appraiser, and the lender We also look at the impact of risk

on interest rates Expected inflation is built into short-term and long-term interestrates, so we also look at the method of extracting expected inflation rates fromthose interest rates

Leases

The chapter on leases gives the reader a picture of the importance of the lease

in a real estate investment Different types of rent are discussed including grossrent, net rent, percentage rent, overage rent, average rent, and effective rent Thefinal two that are listed, average rent and effective rent, are calculated in differentlease scenarios There are differences in how the rent is charged in residentialproperty, office property, warehouse property, and retail property The conceptsand calculations of leasehold value and leased fee value are also addressed Youhave no real estate investment until you have tenants sign a lease and start payingrent, so although this chapter is the last in the book, it could very easily be thefirst chapter

The Financial Calculator

From Chapter 6 through to the end of the book, we are very dependent on the use

of the financial calculator When I tell my students in class that the world revolvesaround the time value of money, they question my sanity As you will see fromthe latter half of the book, much of what goes on in the finance and real estateworlds is closely tied to this concept Don’t worry if you are uncomfortable withthe calculator now You will be close friends with it soon

Review of Math Skills

Math? I Took Math Once

Operations

As a quick review of your very first math course, there are four basic operations in

math calculations: addition, subtraction, multiplication, and division The author

is confident that you are familiar with these processes, which are used repeatedly

in this book Knowing when to add, subtract, multiply, and divide is one of the

keys to successful real estate math calculations

7

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CHAPTER 1 Review of Math Skills

8

Units of Measure

The units of measure used in real estate math calculations can be varied Lotsizes and building sizes use feet, or more precisely, square feet as a basic unit ofmeasure If we take a storage shed measuring 6 feet by 6 feet, its area would be

6 feet× 6 feet = 36 feet2or 36 square feet Notice that both the 6 is squared andthe unit of measure, or feet, is squared You will see both lot sizes and buildingsizes described in number of square feet, which is a measure of size for the area

of a lot or building

Some basic units of measure that are used in real estate include:

1 foot= 12 inches

1 yard= 3 feet

1 acre= 43,560 square feet

1 section= 640 acres = 1 square mile

When measuring volume, the unit of measure is not square feet, square inches,

or square yards, but rather it is cubic feet, cubic inches, or cubic yards That isbecause in order to calculate the volume of a three-dimensional object, you takethe length times the width times the height If our cube is 6 inches by 6 inches by

6 inches, the calculation becomes 6 inches× 6 inches × 6 inches = 216 inches3

or 216 cubic inches

Other units of measure that we may come across in real estate would includeacres One acre is equal to 43,560 square feet Regarding the government surveysystem of legal descriptions, one township is made up of 36 sections of land Onesection is 1 square mile and is made up of 640 acres of land If that is the case,how many square feet are there in one section?

Equation 1.1

640× 43,560 square feet = 27,878,400 square feet

So there are 27,878,400 square feet in one section of land Then how manysquare feet are there in a township?

Equation 1.2

36× 640 × 43,560 square feet = 1,003,622,400 square feet

It appears that there are over 1 billion square feet in one township

Equations

An equation is a mathematical statement that includes an equal sign The values

on either side of the equal sign must be equivalent to each other If they are not

equivalent, then the statement is not true The following statement is not true:

35= 5 × 6 That is, 35 is not the same as 5 × 6 When you take 5 × 6, you do not

get an answer of 35; you get an answer of 30 In order to make the statement true,

you would need to write 30 = 5 × 6, or you could write 35 = 5 × 7 In either

case, the statement being made is true, so they are both valid equations Thirty is

equivalent to 5× 6 and 35 is equivalent to 5 × 7

Equations are the tools that are used to solve problems The problem may be

a long complex word problem, or it could be a short straightforward calculation

In either case an equation can be used to solve the question at hand

In algebra we learned that an equation can be altered, and as long as we do

the same thing to both sides of the equation, it will still be an equation If we are

using the equation from the first part of this section:

Equation 1.3

35= 5 × 7and we divide both sides of the equation by 5, the result is:

Equation 1.4

35/5 = (5 × 7)/5, which says, 7 = 7

so our equation is still a true statement

Some of these types of problems are addressed in this book If algebra scares

you, don’t worry That word will not be mentioned again … too much

ORDER OF OPERATIONS

When there is a problem that requires more than one calculation, the order of

operations can be dictated through the use of parentheses The calculation within

CHAPTER 1 Review of Math Skills

Equation 1.6

7× 5 + 5 = 70the calculation may progress in a different manner If we add the 5 + 5 first,

we get 10 And if we then take the 10 × 7, we get an answer of 70 As youcan see, the parentheses are very important because they indicate which process,

or calculation, should be performed first Without the parentheses a completelydifferent answer would result (in the example, 40 instead of 70)

EQUATIONS FOR AREA AND VOLUME

The calculations of area and volume are referred to earlier in the chapter Thereare specific equations to use in the calculation of each of these items The equationfor finding the area of a rectangle or square is shown in Equation 1.7

Equation 1.7

Area= L × W

In Equation 1.7, L represents the length of one side of the rectangle, and

W represents the width of the rectangle To obtain the area of the rectangle, simplymultiply the length times the width (See Figure 1.1.) As mentioned previously, if

we are finding the area of a square, the length and width are the same, so the areacan be obtained by taking one side times itself, or the side squared If the unit ofmeasure of our rectangle or square is feet, then the answer to our calculation isgiven in square feet If a lot measures 40 feet by 150 feet, the area of that lot isshown in Equation 1.8

If the unit of measure in our calculation is feet, then our answer is given in

cubic feet If a shed is square in shape and is a single floor with measurements

of 6 feet by 6 feet and it has a 6-foot-high ceiling, the volume calculation would

look like Equation 1.10 (see Figure 1.2.)

Equation 1.10

Volume= 6 feet × 6 feet × 6 feet = 216 cubic feet

The area of a circle is only somewhat more complex in that we must use a

number called pi, or the Greek symbolπ Pi (π) is equal to approximately 3.14

The area of a circle is as follows:

Equation 1.11

Area=π(r2)

CHAPTER 1 Review of Math Skills

12

Figure 1.2 Cube

wherer is equal to the radius of the circle andπ, or pi, is equal to 3.14 rounded

to two decimal places (See Figure 1.3.) The radius is the distance from the center

of the circle to the outside of the circle itself, or one half of the diameter of thecircle If we wanted to find the area of a circle that has a radius of 20 feet, thecalculation would be as follows

Equation 1.12

Area= 3.14(202) = 3.14(400) = 1,256 square feet

The area of a circle does not come up too often in real estate, but it may arise

in an agricultural setting where the square footage covered by an irrigation systemmust be found

Figure 1.3 Circle

Figure 1.4 Cylinder

The volume of a cylindrical tube, like a silo, would be the area of the base

times the height (See Figure 1.4.) The calculation would look like this:

Equation 1.13

Volume=π(r2) × H

It is simply the area of the circular base times the height If a silo is 35 feet

from side to side (diameter = 35 feet) and stands 25 feet tall, what would its

volume be? The calculation would look like this:

Equation 1.14

Volume= 3.14(35/2)2× 25 = 3.14(17.5)2× 25 = 3.14 × 306.25 × 25

= 24,040.63 cubic feet

Recall that the calculation of the area of a circle calls for the radius squared

timesπ, and in the preceding problem we were given the diameter As a result,

in Equation 1.14 we needed to take the diameter, 35, and divide it by 2 to get the

radius of 17.5

A trapezoid is a four-sided figure in which the corners are not all right angles

(90 degrees) and the bases are not the same length but they are parallel to

each other (see Figure 1.5) The calculation for the area of a trapezoid is as

CHAPTER 1 Review of Math Skills

14

Figure 1.5 Trapezoid

20 feet and 30 feet and a height of 10 feet would be:

Equation 1.16

Area=1/2(20 + 30) × 10 =1/2× 50 × 10 = 250 square feet

The Financial Calculator

Starting with Chapter 6, “Time Value of Money,” this book depends strongly

on the use of the financial calculator This time-saving device has replaced thecumbersome time value of money tables that had been used in the past In real

estate, the tables were referred to as the six functions of a dollar The subtitle

of Chapter 6 makes reference to the equations that made use of the tables, “SixEquations That Will Change Your Life.” The calculator solutions that are used inthe book have reduced the dependence on the equations However, for someonewho truly wants to understand time value of money, understanding the equations

is a prerequisite

THE TI BUSINESS ANALYST II PLUS CALCULATOR

A number of the calculations that are addressed in this book are simplified throughthe use of a financial calculator Chapter 6 depends extensively on this calculator.Subsequent chapters also make much use of the financial calculator While thecalculations can be made using a standard four-function calculator and a set ofpresent value/future value tables, the financial calculator makes the job mucheasier In those chapters that make use of the financial calculator, you will seethe solutions to the problems spelled out in calculator keystrokes The calculatorthat is used is the Texas Instruments Business Analyst II Plus financial calculator

It is probably the most cost-effective financial calculator in use today It is readily

available in office supply stores and college bookstores Some of the important

keys for us are described in the following section

IMPORTANT CALCULATOR KEYS

The BA II Plus calculator has many abilities that we do not address in this book

The ones we primarily focus on are the time value of money keys and the cash

flow keys The time value of money keys are in the third row from the top of the

calculator and are as follows: [N] [I/Y] [PV] [PMT] [FV] The [N] key stands for

the number of compounding periods in the calculation The [I/Y] key is the interest

Figure 1.6 The Texas Instruments BA II Plus calculator

Texas Instruments BA II PlusTMimage used with permission

CHAPTER 1 Review of Math Skills

16

rate per year The [PV] represents the present value in a problem The [PMT] isthe payment in an annuity calculation, and finally the [FV] is the future valuecalculation These keys are put into context in the time value of money (TVM)discussion in Chapter 6

Other keys that we make use of are in the first row These include the [cpt]key, which tells the calculator to compute a value The [enter] key is used to enter

a value The up and down arrow are used to move up and down within a cashflow worksheet that we use in the real estate investment chapter (Chapter 11) The[on/off] key is fairly self-explanatory

The second row of keys is also used in the investment chapter as well as others.The [2nd] key moves us up to the second level of many of the keys You will notice

in the photo of the calculator in Figure 1.6 that the second level of the [FV] keysays CLR TVM If we hit the [2nd] key followed by the [FV] key, that clears outthe TVM registers In other words that process puts zeros in those TVM registers

so that we can start from scratch on a new problem If you don’t clear out thoseregisters and you move from one type of problem to another, an old number fromthe previous problem will try and work its way into the new problem and end upgiving you a bogus answer The [CF] key stands for cash flows, and we use thatkey to get into the cash flow worksheet in our investment analysis The [NPV]key is used for calculating the net present value of a series of cash flows The[IRR] is used when we want to calculate the internal rate of return of a series of

cash flows (The terms NPV and IRR are defined further in Chapter 11.)

The worksheet can be cleared after an NPV problem or an IRR problem by ting the [2nd] key followed by the [CLR WORK] key in the lower left-hand corner

hit-of the calculator Additional keys are described as their use becomes necessary

Quiz for Chapter 1

1 The home you are selling has a rectangular lot with dimensions of 75 feet

by 150 feet What is the area of the lot?

a 10,750 square feet

b 12,000 square feet

c 11,250 square feet

d 11,750 square feet

2 Your driveway is 20 feet by 20 feet How many square feet of pavement

will you have if you have the driveway tarred?

a 400 square feet

b 4,000 square feet

c 600 square feet

d 400 cubic feet

3 If your garage has a flat roof, what is its volume if its dimensions are

22 feet by 24 feet and it stands 8 feet tall?

5 Your neighbor’s lot is trapezoidal in shape The two bases are 50 feet

and 80 feet in length The perpendicular distance between the bases is

100 feet What is the area, in square feet, of your neighbor’s lot?

a 400,000 square feet

b 13,000 square feet

c 6,500 square feet

d 13,000 cubic feet

6 You are planning to make a skating rink in your back yard You put a

stake in the center of the yard and by using a rope you measure a circle

CHAPTER 1 Review of Math Skills

a 1,306 feet

b 156 feet

c 166 feet

d 126 feet

10 The lot you recently sold had a sale price of $65,000 The lot had a

front footage of 80 feet and a depth of 165 feet What was the price per

square foot it sold for and the price per front foot?

a $4.92 and $393.94

b $393.94 and $812.50

c $80.00 and $393.94

d $4.92 and $812.50

CHAPTER

Fractions, Decimals, and Percentages

Which One of These Is Not Like

the Others?

A large portion of math is involved with calculating and manipulating parts of

a whole These parts may be fractions, they may be decimals, or they may

be percentages Real estate is no different You may be dealing with 1/2 of anacre, or an operating expense ratio of 0.35, or calculating 6 percent of the saleprice In each case we are dealing with a part of a whole something The greater

20

Copyright © 2007 by McGraw-Hill, Inc Click here for terms of use

your ease in moving from one measuring unit to another, the greater is your chance

of success in working in real estate

Fractions

There are a couple of different ways to look at a fraction, and the best way

for you to look at a fraction is the way that makes the most sense to you The

first way to look at a simple fraction (a fraction with a value of less than 1) is

by analyzing both the numerator (top number) and the denominator (bottom or

“down” number) The denominator tells us how many parts our whole is divided

into If the fraction is3/4, that tells us that our whole item is divided into four

parts The numerator tells us how many of those parts we are actually dealing

with Again if our fraction is 3/4, the numerator of 3 tells us that we are only

dealing with 3 of those 4 parts that the whole is divided into If the fraction is1/2,

it says that our whole item is divided into 2 parts and that we are currently looking

at only 1 of those parts If the fraction is2/2, then the item is divided into two

parts, and we are looking at both of them So if the fraction is2/2we are looking

at 2 of 2 parts, in other words we are looking at the whole item, so 2/2 = 1

Whenever the numerator and denominator are the same, the value of the fraction

is the whole number 1 Thus, the value of each of these fractions is 1:2/2,3/3,5/5

and1,000/1,000

MIXED NUMBERS

Another way to look at a fraction is that a fraction is a division problem waiting to

happen If the numerator is greater than the denominator (or the value is greater

than 1), the solution or answer to the division problem will be a mixed number

By performing the operation that is called for, we are beginning to manipulate a

fraction In Equation 2.1,

Equation 2.1

5/4= 5 ÷ 4 = 11/4

you divide the 5 by 4, and a you get the whole number 1, plus a remainder of 1

which becomes the numerator, and 4 is your denominator, so the answer is 11/4

This is referred to as a mixed number, because there is a whole number and a

fraction If we want to convert this back to a fraction, we take the denominator, 4,

CHAPTER 2 Fractions, Decimals, Percentages

22

times the whole number, 1, and add the numerator 1 and we get 5/4 If we takethe fraction7/2, we could convert that to a mixed number by dividing 7 by 2 andget the answer 3 plus a remainder of 1 (which becomes the numerator over thedenominator of 2) to give us the mixed number 31/2 That 31/2 in turn could beconverted to a mixed number by taking the denominator of 2 times the number 3

to get 6, plus the numerator of 1 gives us7/2

REDUCING FRACTIONS

A fraction can be reduced to simpler form The fraction 4/8 can be reduced to

2/4, and 2/4 can be reduced to 1/2 For the fraction 4/8 divide the numerator by

2 and the denominator by 2, and that yields an answer of 2/4 Do the same

to 2/4, that is, divide both the numerator and denominator by 2 and get theanswer 1/2 Alternatively, just dividing the numerator and denominator of 4/8

by 4 will also give the answer of 1/2 This is called reducing a fraction to its

simplest form.

MANIPULATING FRACTIONS

Adding fractions is just like adding whole numbers In order to add 1/4 plus

3/4, simply add the numerators, 1+ 3 = 4, and put the answer back over thedenominator,4/4 In like fashion3/8+7/8 = (3 + 7)/8 =10/8 That answer could

be converted to the mixed number 12/8, which can be reduced to11/4 In similarfashion3/4minus 1/4would result in an answer of2/4, or1/2

The requirement that must be met before adding or subtracting fractions isthat they must first have the same denominator If there are fractions with dif-ferent denominators, they can still be added or subtracted, but they must first

be converted to a common denominator A common denominator is a

denom-inator that the two fractions would have in common with each other, given

a little manipulation The simplest way to find a common denominator would

be to multiply together the two denominators For example, to add the twofractions 1/4 + 1/3, multiply the 4 times the 3 and get a common denomi-nator of 12 In effect we multiply each fraction by 1 In the case of thefraction 1/4, we will multiply it by 3/3 to get an answer of 3/12 We multi-ply the 1/3 by 4/4 to get an answer of 4/12 The result is 1/4 +1/3 = 3/12 +

4/12 = 7/12

Following the steps from the previous paragraph, take a minute and add the

fractions1/2+3/7 Let’s see what you might have done

Equation 2.2

1/2+3/7= (7/7×1/2) + (2/2×3/7) =7/14+6/14=13/14

The common denominator for the two fractions would be 2× 7 = 14 Multiply

the 1/2 by 7/7 and get 7/14 Then multiply 3/7 by 2/2 and get 6/14 Now add the

7/14+6/14and get13/14as an answer By multiplying1/2by7/7and3/7by2/2, we

multiplied each fraction by 1 and any number multiplied by 1 gives us the same

number Just like 1× 5 = 5,1/2×7/7=1/2, only this time we are calling1/2by

the name7/14so that we can add it to6/14

If this didn’t make sense when you read it the first time, read it over again

and jot down the fractions on a sheet of paper as you go There is no penalty

associated with reading some material twice

Converting Fractions to Decimals

The division problem that is being called for in a fraction might not deal with

a remainder, as in the explanation of Equation 2.1 above Rather, it may make

use of a decimal point, and the answer may be in decimal form Let’s look at the

fraction3/4 again The fraction asks us to divide 3 by 4 If we were to do this in

long division, or more likely today, punch the numbers into a calculator, 3÷ 4,

the answer to the calculation is 0.75 Thus 0.75 is the decimal equivalent of3/4, or

3/4 = 0.75 (See Chapter 1 for the calculator key strokes for the TI Professional

Business Analyst calculator.)

Your calculator key strokes should look like this for the TI calculator:

Equation 2.3

3[÷] 4 [=] 0.75

This is the type of answer you will get when the fraction is a simple fraction

with a numerator that is less than the denominator; in other words it has a value of

less than 1 The answer will begin with a decimal point, again indicating a value

of less than 1 If the division problem has a numerator that is greater than the

denominator, then the answer will begin with a whole number with a decimal point

CHAPTER 2 Fractions, Decimals, Percentages

24

somewhere in the middle of the number with additional zeros or numbers to theright of the decimal point If the fraction is5/4, the TI calculator key strokes wouldlook like this:

If Equation 2.1 is true and Equation 2.4 is true, then it must be true that:

Equation 2.5

5/4= 11/4= 1.25

To verify that, take1/4and punch it into your calculator, 1 [÷] 4 [=] 0.25 plus

4/4which is equal to 1, so 1+ 0.25 = 1.25 So Equation 2.5 is indeed an equationbecause the elements on either side of the equal sign, in this case two equals signs,are indeed equivalent to each other

We have discovered two ways to find the decimal equivalent of5/4 Either take

5 [÷] 4 = 1.25 or take 1 [÷] 4 [=] 0.25 and add it to the 1 to get 1.25

As practice, find the decimal equivalents of each of the fractions in Table 2.1.Cover up the right-hand column as you calculate the decimal equivalent of each

of the following fractions

Table 2.1 Fractions to decimals Fractions Decimals

Converting Decimals to Percentages

and Back

The second decimal place to the right of the decimal point represents hundredths

A percentage is a statement in hundredths of a whole, or 1 hundredth= 1 percent =

0.01 Sixty-five hundredths stated as a decimal is= 0.65 To convert that decimal

value to a percent, simply move the decimal point two places to the right and add a

percent sign, so 0.65= 65% This is again an equation because the value to the left

of the equal sign is equivalent to the value to the right of the equal sign The

dec-imal value may be stated in thousandths rather than hundredths It doesn’t matter;

we still move the decimal places two places to the right to get the percentage

value That being the case, a decimal value of 0.855 is equivalent to 85.5%

The need to convert decimals to percentages arises all the time in different real

estate settings For example, your taxes went up from $3,000 per year to $3,500

per year, and you want to know what percent increase that was The taxes started

at $3,000, and they went up to $3,500 The ending value of $3,500 minus the

beginning value of $3,000 gives us an increase of $500 per year Taking the $500

increase and dividing it by the $3,000 beginning value gives us an increase of

0.167 ($500 ÷ $3,000 = 0.167) Now convert that to a percentage by moving

the decimal point two places to the right and adding a percent sign, and we see

that your taxes went up by 16.7% Alternatively we could take the ending value

($3,500) divided by the beginning value ($3,000), $3,500÷ $3,000 = 1.167 We

then subtract the 1, shift our decimal point two places to the right, add the percent

sign, and we get 16.7% Either method is acceptable

Another situation may involve an investor She has purchased a duplex for

$285,000 and the assessor has told her that the land value is $57,000 She would

like to know what percentage of the total purchase price is represented by the land

value She should take the land value and divide it by the total purchase price,

$57,000 ÷ $285,000 = 0.20 and then move the decimal point two places to the

right and add a percent sign This answer indicates that the land value represents

20% of the total purchase price

Maybe your city planner tells you that the maximum coverage your new

build-ing can have on your lot is 55% accordbuild-ing to the city buildbuild-ing requirements You

know that your lot is a rectangle of 125 feet by 150 feet The building you are

plan-ning to put up has a footprint, or a first floor square footage, of 9,750 square feet

Do you fit within the city’s requirements? First find the area of the lot Take

125 feet× 150 feet and get 18,750 square feet as the area of the lot Then take

the square footage of the first floor of the building and divide it by the area of the

lot; 9,750 ÷ 18,750 = 0.52 = 52% Your building will cover 52% of the lot, so

you are within the city code for area coverage

CHAPTER 2 Fractions, Decimals, Percentages

percent-to the left and drop the percent sign and get 0.06 Now take the 0.06× $355,000and find the gross commission value which is $21,300

Maybe you are working as a property manager and your boss told you thatyou are being given a lot more responsibility and you will now be managing fourbuildings instead of the three that you have been working on in the past As aresult you will receive a raise of 8% You are currently making $42,000 per year.How much will your raise be? How much will you be making next year? Take the8% and shift the decimal point two places to the left, drop the percent sign andget 0.08 Now take the 0.08× $42,000, and your raise is $3,360 Add that $3,360

to the $42,000 to find you will be making $45,360 next year As an alternatemethod take the $42,000 and multiply it times 1.08 You also get $45,360 as yournext year’s salary

In Table 2.3, cover up the right-hand column and calculate the decimal alents of the percentages that are given Once you have finished, uncover theright-hand column, and the correct answers will again magically appear

equiv-Part, Percent, Whole

Many real estate situations call on the use of the part, percent, whole concept

It is an example of simple algebra, but don’t let that word scare you Look atFigure 2.1

Table 2.3 Percent to decimal Percents Decimals

Figure 2.1 Part, percent, whole

You will notice that the part is above the horizontal line and the percent and

whole are below the horizontal line Think of that horizontal line as the line in a

fraction In effect we are creating two different fractions and one multiplication

problem The first fraction illustrated is part/whole To make this an equation, we

simply plug in the percent, as shown in Equation 2.6

Equation 2.6

Part/whole = Percent

This tells us that if we take a part of the whole and divide by the total whole,

it will give us the percentage of the whole that the part represents Let’s say

that we have a whole sale price of $125,000 and we know that the commission

that is being paid on the transaction is $7,500 The question is, what was the

commission rate on the deal? We simply plug our number into Equation 2.6 and

get $7,500/$125,000 and get 0.06 or 6%

The second equation that Figure 2.1 is begging us to create is Equation 2.7

Equation 2.7

Part/percent = Whole

This equation tells us that if we take the part and divide it by the percentage

that it represents, the resulting answer will be the whole that we are dealing

with If our net proceeds from the sale are $134,850 and that is after paying

CHAPTER 2 Fractions, Decimals, Percentages

28

a 7% commission, then the net proceeds actually represent 93% of the whole, so

we take $134,850/0.93 and get a whole sale price of $145,000 If, on the otherhand, we had been given the commission amount of $10,150 and had been toldthat it was 7% of the sale price, we would have taken $10,150/0.07 and gottenthe same whole sale price of $145,000

The third equation that is being called for in Figure 2.1 is multiplying the twoitems below the line to get the value above the line

Equation 2.8

Percent× whole = PartThis calculation is probably used more often than the other two This time wecan take our commission rate times our sale price and get the dollar amount ofthe commission Given a sale price of $325,000 and a commission rate of 6%,this should result in a commission value of $325,000× 0.06 = $19,500

You will find a variation on the part, percent, whole equations used extensively

in Chapter 10 on real estate appraisal The equations in appraisal are referred to

as the IRV formula, where I is equal to the NOI (income) of the property, R is thecapitalization rate and V is the value of the property The resulting equations areI/R= V, or income over rate equals value; I/V = R, or income over value equalsrate; and R× V = I, or rate times value equals income

Percentages and Mortgages

Mortgage amounts are typically stated in terms of loan-to-value ratios A ratio is

one number divided by another In the loan-to-value ratio, the loan or mortgageamount is divided by the value of the property, like this—(loan ÷ value) or(loan/value)—and the answer is given as a percent If we are purchasing a homefor $100,000 and we are getting an $80,000 mortgage, our loan-to-value ratio is

$80,000/$100,000= 0.80 = 80% In similar fashion only this time working in theother direction, if we are purchasing a home for $145,000 and we are getting a75% loan to value mortgage, $145,000× 0.75 = $108,750, we will be borrowing

$108,750 in the form of a mortgage loan

One minus the loan-to-value ratio is the equity-to-value ratio If you think ofthe total value of the home as 1, or 100%, and your loan to value ratio is 80%, thenyour equity-to-value ratio must be 20%, 100% – 80%= 20% This emphasizesthe concept that the market value of the property is made up of equity and debt,where debt is the mortgage on the home The difference between the market value

of the home and the mortgage balance is your equity, as stated in Equation 2.9

When you just purchase the home, your equity is the same as your down payment,

and your mortgage balance is the amount you just borrowed from the bank in the

form of a home mortgage

Equation 2.9

Equity= Market value − mortgage balance

As time goes by, your equity increases because of two changes (see

Equa-tion 2.10) First of all, your equity increases because of any appreciaEqua-tion that

takes place in the value of your home This increase in market value gives you

more equity Second, your equity increases because the principal balance on your

mortgage decreases over time because the principal is reduced each time you make

a mortgage payment (See Chapter 7.)

Equation 2.10

↑ Equity = ↑ Market value − ↓ mortgage balanceEquation 2.10 states that an increase in equity can come about as a result of an

increase in market value and/or a decrease in mortgage balance As long as your

housing market experiences no housing price bubble, the first caveat will be true

And as long as you have an amortized mortgage and you continue to make your

mortgage payments, the second caveat will be true

CHAPTER 2 Fractions, Decimals, Percentages

30

Quiz for Chapter 2

1 Add the following sets of fractions Where necessary, find the commondenominators before adding Reduce the answers to their simplest formwhere necessary

e 21/4= ?

f 43/4= ?

6 List the percentages represented by those same fractions and mixed

numbers in question 5

7 Sylvia purchased a home earlier this year for $167,750 and took out a

mortgage for $142,375 at 8% for 30 years What was the loan-to-value

ratio on the mortgage, and what was her down payment as a percentage

of the purchase price?

9 Your brother Jake just sold his house for $220,000 and netted $206,800

on the sale What was the commission rate that he paid his broker?

a 5%

b 6%

c 7%

d 7.5%

10 Jake, in question 9, had purchased his house only one year earlier He

purchased the house for $215,686 What rate of appreciation had he

earned in that one year?

a 5%

b 4%

c 3%

d 2%